Invariant Domains and First-Order Continuous Finite Element Approximation for Hyperbolic Systems

We propose a numerical method to solve general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on non-uniform grids. The properties of the method are based on the introduction of an artificial dissipation that is defined so that any convex invariant sets containing the initial data is an invariant domain for the method. The invariant domain property is proved for any hyperbolic system provided a CFL condition holds. The solution is also shown to satisfy a discrete entropy inequality for every admissible entropy of the system. The method is formally first-order accurate in space and can be made high-order in time by using Strong Stability Preserving algorithms. This technique extends to continuous finite elements the work of \cite{Hoff_1979,Hoff_1985}, and \cite{Frid_2001}

Kinetic schemes on staggered grids for barotropic Euler models: entropy-stability analysis

International audienceWe introduce, in the one-dimensional framework, a new scheme of finite volume type for barotropic Euler equations. The numerical unknowns, namely densities and velocities, are defined on staggered grids. The numerical fluxes are defined by using the framework of kinetic schemes. We can consider general (convex) pressure laws. We justify that the density remains non negative and the total physical entropy does not increase, under suitable stability conditions. Performances of the scheme are illustrated through a set of numerical experiments